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Cluster structures for the A singularity

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  • Jenny August
  • ,
  • Man Wai Cheung, University of Tokyo
  • ,
  • Eleonore Faber, University of Leeds
  • ,
  • Sira Gratz
  • Sibylle Schroll, University of Cologne, Norwegian University of Science and Technology

We study a category (Formula presented.) of (Formula presented.) -graded maximal Cohen-Macaulay (MCM) modules over the (Formula presented.) curve singularity and demonstrate that it has infinite type (Formula presented.) cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type (Formula presented.) cluster categories of Holm–Jørgensen, Fisher and Paquette–Yıldırım. As a consequence, (Formula presented.) has cluster tilting subcategories modelled by certain triangulations of the (completed) (Formula presented.) -gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed (Formula presented.) -gon.

Original languageEnglish
JournalJournal of the London Mathematical Society
Pages (from-to)2121-2149
Publication statusPublished - Jun 2023

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© 2023 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

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